Advertisements
Advertisements
рдкреНрд░рд╢реНрди
Differentiate the function with respect to x.
`x^(xcosx) + (x^2 + 1)/(x^2 -1)`
Advertisements
рдЙрддреНрддрд░
Let, y = `x^(x cos x) + (x^2 + 1)/(x^2 - 1)`
Again, let y = u + v
Differentiating both sides with respect to x,
`dy/dx = (du)/dx + (dv)/dx` ...(1)
Now, u = `x^(x cos x)`
Taking logarithm of both sides,
log u = `log x^(x cos x)`
log u = x cos x log x
Differentiating both sides with respect to x,
`1/u (du)/dx = x cos x d/dx log x + log x d/dx x cos x`
= `x cos x * 1/x + log x [x d/dx cos x + cos x d/dx (x)]`
= cos x + log x [x (−sin x) + cos x]
= cos x + x (−sin x) · log x + cos x · log x
`therefore (du)/dx = u [cos x log x - x sin x log x + cos x]`
= `x^(x cos x)` [cos x log x − x sin x log x + cos x] ....(2)
Also, v = `(x^2 + 1)/(x^2 - 1)`
Differentiating both sides with respect to x,
`(dv)/dx = ((x^2 - 1) d/dx (x^2 + 1) - (x^2 + 1) d/dx(x^2 - 1))/((x^2 - 1)^2)`
= `((x^2 - 1)(2 x) - (x^2 + 1) (2 x))/((x^2 - 1)^2)`
= `(2 x [x^2 - 1 - x^2 - 1])/((x^2 - 1)^2)`
= `(-4x)/((x^2 - 1)^2)` ....(3)
Putting the values тАЛтАЛof from equation (2) and (3) in equation (1),
`therefore dy/dx = (du)/dx + (dv)/dx`
`= x^(x cos x) [cos x log x - x sin x log x + cos x] - (4x)/(x^2 - 1)^2`
APPEARS IN
рд╕рдВрдмрдВрдзрд┐рдд рдкреНрд░рд╢реНтАНрди
Differentiate the following function with respect to x: `(log x)^x+x^(logx)`
if xx+xy+yx=ab, then find `dy/dx`.
Differentiate the function with respect to x.
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
Differentiate the function with respect to x.
(log x)cos x
Differentiate the function with respect to x.
xx − 2sin x
Differentiate the function with respect to x.
xsin x + (sin x)cos x
Find `bb(dy/dx)` for the given function:
yx = xy
Find `bb(dy/dx)` for the given function:
(cos x)y = (cos y)x
If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.
If y = `e^(acos^(-1)x)`, −1 ≤ x ≤ 1, show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`.
If `y = sin^-1 x + cos^-1 x , "find" dy/dx`
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
xy = ex-y, then show that `"dy"/"dx" = ("log x")/("1 + log x")^2`
If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`
If log (x + y) = log(xy) + p, where p is a constant, then prove that `"dy"/"dx" = (-y^2)/(x^2)`.
`"If" y = sqrt(logx + sqrt(log x + sqrt(log x + ... ∞))), "then show that" dy/dx = (1)/(x(2y - 1).`
If ey = yx, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.
Find the second order derivatives of the following : x3.logx
Find the second order derivatives of the following : log(logx)
If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.
If y = A cos (log x) + B sin (log x), show that x2y2 + xy1 + y = 0.
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
If y = 5x. x5. xx. 55 , find `("d"y)/("d"x)`
The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290 K and the metal temperature drops from 370 K to 330 K in 1 O min, then the time required to drop the temperature upto 295 K.
lf y = `2^(x^(2^(x^(...∞))))`, then x(1 - y logx logy)`dy/dx` = ______
`d/dx(x^{sinx})` = ______
If `("f"(x))/(log (sec x)) "dx"` = log(log sec x) + c, then f(x) = ______.
If y = `("e"^"2x" sin x)/(x cos x), "then" "dy"/"dx" = ?`
`2^(cos^(2_x)`
If `f(x) = log [e^x ((3 - x)/(3 + x))^(1/3)]`, then `f^'(1)` is equal to
If `log_10 ((x^2 - y^2)/(x^2 + y^2))` = 2, then `dy/dx` is equal to ______.
If xy = yx, then find `dy/dx`
If \[y=x^x+x^{\frac{1}{x}}\] then \[\frac{\mathrm{d}y}{\mathrm{d}x}\] is equal to
